Optimal Liquidation Trajectories for theAlmgren-Chriss Model with Lévy Processes
Arne Lokka(joint work with Junwei Xu)
London School of Economics
a.lokka@lse.ac.uk

Paris, September 2016

Overview

1 Introduction
2 Lévy Models and Optimal Liquidation Trajectories
3 Linear Approximation of Exponential Lévy Models
4 Properties of the Optimal Trajectories and the Temporary

Impact Function
5 Conclusion and References

Introduction
The Almgren-Chriss model
Let Yt denote the number of shares the investor
R t holds at time t,which is assumed to take the form Yt = Y00 ⇠u du. Then theinvestor sells at prices S given by
St = s + Wt + ↵ Y tY0 )F (⇠t ),

t0.

The temporary impact function F is of the form F (x) = x ,where typically is around 0.6.
The objective for the investor is to maximize
⇥ ⇤
E CTY
Var CTY ,where CTY is the cash position at time T corresponding to theliquidation strategy Y .

Introduction
This talk
We consider an infinite time-horizon where the investor sells atprices given by
St = s + Lt + ↵ YtY0 )F (⇠t ),

t0,

where Lt is a Lévy process and F is a function satisfying certain
properties.
The objective for the investor is to find a strategy Y whichmaximize⇥⇤YE exp AC1,Y denotes thewhere A denotes the investor’s risk aversion and C1
investor’s cash position at the end of time.

Initial Observations
Observation 1
In the Brownian motion case, the optimal solution to the problemof maximising the expected exponential utility of the cash positionis equal to the solution to the problem of maximising themean-variance criterion over deterministic strategies (the optimalliquidation trajectories coincide).

Initial Observations
Observation 1
In the Brownian motion case, the optimal solution to the problemof maximising the expected exponential utility of the cash positionis equal to the solution to the problem of maximising themean-variance criterion over deterministic strategies (the optimalliquidation trajectories coincide).

Observation 2
For small time-horizons, exponential Lévy models provide a good
fit to observed stock price data (e.g. exponential variance-gamma
or NIG models), and liquidation is normally completed within short
time-horizons.

Initial Observations
Observation 1
In the Brownian motion case, the optimal solution to the problemof maximising the expected exponential utility of the cash positionis equal to the solution to the problem of maximising themean-variance criterion over deterministic strategies (the optimalliquidation trajectories coincide).

Observation 2
For small time-horizons, exponential Lévy models provide a good
fit to observed stock price data (e.g. exponential variance-gamma
or NIG models), and liquidation is normally completed within short
time-horizons.

Observation 3
Models based on Brownian motion underestimate the probability oflarge market movements.

Some Key Questions
Question 1
What conditions do we have to impose on the temporary impact
function F ?

Some Key Questions
Question 1
What conditions do we have to impose on the temporary impact
function F ?

Question 2
What is our set of admissible strategies?

Some Key Questions
Question 1
What conditions do we have to impose on the temporary impact
function F ?

Question 2
What is our set of admissible strategies?

Question 3
What are the properties of the optimal liquidation trajectories?

Some Key Questions
Question 1
What conditions do we have to impose on the temporary impact
function F ?

Question 2
What is our set of admissible strategies?

Question 3
What are the properties of the optimal liquidation trajectories?

Question 4
Are there any natural relations between the temporary impact
function F and the probability law of the underlying asset?

Model Description
Assumption
We assume that L is a non-trivial Lévy process such that L1 has
finite second moment and⇥⇤¯ = inf < 0 | E e L1 < 1 < 0.
Such a Lévy process admits a decomposition
Ze dx),
Lt = µt + Wt +x N(t,Rt0,

where µ 2 R and0 are constants, W is a standard Brownian
motion, Ne is a compensated Poisson random measure. We let ⌫denote the Lévy measure associated with L.

Model Description
Admissible strategies
Given an initial share position y , the set A(y ) of admissible
strategies consists of adapted, absolutely continuous,
Rt
non-increasing processes Y of the form Yt = y0 ⇠u du,satisfyingZ 1kYt kL1 (P) dt < 1
if µ 6= 0,(1)0andZ 10kYt k2L1 (P) dt < 1

if µ = 0.

(2)

We let AD (y ) denote the set of all deterministic strategies in A(y ).

Model Description
The investor sells his shares at prices given by
St = s + Lt + ↵ Yt
where ↵Y0 )F (⇠t ),

t0,0 and F : [0, 1) ! [0, 1) satisfies(i) F is continuous and continuously di↵erentiable on (0, 1);(ii) F (0) = 0;(iii) the function x 7! xF (x) is strictly convex on [0, 1);(iv) limx!0 xF 0 (x) exists;(v) the function x 7! x 2 F 0 (x) is strictly increasing on [0, 1), andtends to +1 as x ! 1.

The Cash Position
For an admissible liquidation strategy Y 2 A(y ), the investor’s
cash position at the end of time isZ 1
YC1 = cSt dYt0Z 1Z 11 2
= c + sy↵y +Yt dLt⇠t F (⇠t ) dt.
200Also, recall that the investor’s optimisation problem is to find aY ⇤ 2 A(y ), which maximise⇥⇤YE exp AC1.

Problem Simplification
It turns out that in order to solve the optimal liquidation problem,
it is sufficient to solve (the derivation is similar to that of Schied,
Schöneborn and Tehranchi (2010))
Z 1⇢
V (y ) = inf
A (Yt ) + A⇠t F (⇠t ) dt,
Y 2AD (y ) 0

where⇥A (y ) = ln E exp⇤
AyL1 .

Moreover, if µ > 0, then there are no admissible optimal solutions.
Hence from now on we assume that the asset has a drift µ  0.

The Optimal Liquidation Trajectory
Let G : [0, 1) ! [0, 1) be the inverse function of x 7! x 2 F 0 (x),and define
Z y1⌧=du,y 2 [0, ¯A ).
A (u)
0 G
ALet Y ⇤ be the unique process satisfying
Z y1
du = t, if t  ⌧,and
A (u)Yt⇤ GAYt⇤ = 0, if t > ⌧.

Then Y ⇤ 2 AD (y ) is the optimal liquidation strategy, and theassociated speed ⇠ ⇤ is given by✓◆A (Yt⇤ )⇤⇠t = G,0  t < ⌧.
A

Approximation of Exponential Models
Stock price data are typically calibrated to exponential Lévymodels, so we want to derive a linear approximation of suchmodels. Consider the exponential model
Set = s exp Let + It ,where It = ↵(Yt Y0 ) + F (⇠t ) is the price impact at time t, and Le
is a Lévy process withRcharacteristics (eµ, e, ⌫e) satisfyingd ⌫e(x) = fe(x) dx and |z| 1 e2z ⌫e(dz) < 1. We want to choose aLévy process L such that S given by
St = s + L t + I teis the linear approximation of S.

Approximation of Exponential Models
Solution
Define a measure ⌫ by⌫(dx) =and set⌘1 e⇣f ln(1 + x) dx,1+xm=µe+e2+2ZR(ez1x>

1, x 6= 0,z1{|z|<1} ) ⌫e(dz).

Let L be a Lévy process with characteristics (m, e, ⌫). Then
St = s + sLt + Ite
is the linear approximation of S.

Example; Variance-GammaConsider the case where Le is the variance-gamma process withparameters (✓, ⇢, ⌘). Then e = 0, and ⌫e has a density
where1 Cz D|z|fe(z) =e,⌘|z|C=✓⇢2andD=z 2 R,q2
✓2 + 2⇢⌘⇢2.

Our assumptions are satisfied if D C > 2, and the cumulant
generating function e admits the expression✓◆
1x 2 ⇢2 ⌘e(x) =ln 1
✓⌘x .
⌘2

Example; Variance-GammaThe Lévy measure for the process L appearing in the expression forthe linear approximation of Se in the VG case is(1C +D 1 dx,
x 2 ( 1, 0),⌘ ln(1+x) (1 + x)⌫(dx) =1CD
1dx,x 2 (0, 1).
⌘ ln(1+x) (1 + x)
The corresponding A function isZ 1⇣vg
A (u) = Amu +e Aux1

where m = e(1).

⌘1 + Aux ⌫(dx),Examples; Power-Law Temporary Impact
In this case, the temporary impact function takes the form
F (x) = x .
For our numerical examples, we choose = 0.6 and= 4.7 ⇥ 10 5 (which we believe are reasonable in view of
Almgren et. al. (2005) when we work with a daily volatility ofroughly 0.02, take daily volume to be 2 ⇥ 106 and choose an initialstock price of s = 100). The function G then takes the form
G (x) =✓x◆ 1+1,x0,

and the optimal liquidation trajectory is given by⇠t⇤ =✓ vg ⇤ ◆ 1
sA (Yt ) +1.
AExample; Power-Law Temporary Impact
We choose a risk aversion of A = 2, and recall that the una↵ected
price process is given by 100 ⇥ exp Let .

Exponential VG caseWe choose Le to be a VG Lévy process with parameters✓ = 0.002, ⇢ = 0.02 and ⌘ = 0.6, which are typical values fordaily stock price data.

Exponential BM caseIn this case we choose Let = µt + Wt , where W denotes astandard BM. We let the parameter µ and be given by2
2µ+ 2 =e(1) and 2µ + 42 = e(2), in which case the VG and theBM models have the same mean and variance.

Initial position

We assume that the investor initially holds y = 2 ⇥ 105 number ofshares, which is 10% of daily volume.

Optimal Liquidation Trajectory BM-Case
Yt
200 000

150 000

100 000

50 0000.00005

0.000100.000150.00020t0.00025

If the investor follows the optimal liquidation strategy, it takes
roughly 2 ⇥ 10 4 days until the time the investor has liquidated
99% of his shares.

Optimal Liquidation Trajectory VG-Case

If the investor follows the optimal liquidation strategy, it takes lessthan 2 ⇥ 10 108263 days until the time he has liquidated 99% of hisshares.
So essentially, the investor liquidates 99% of his shares instantly.
Even if we take = 1000, the time it takes the investor toliquidate 99% of a position of 2 ⇥ 105 number of shares would beless than 2 ⇥ 10 168 .

Optimal Liquidation Trajectory VG-Case

The previous example was an extreme case with risk aversion
A = 2. Almgren typically assumes a risk aversion in the region
A = 104 to A = 105 .
For risk aversions in the region A = 105 , the BM and the VGmodels will produce liquidation strategies that are comparable.

An Equivalence Relation
Consider the VG model with temporary impact function Fvg andthe BM model with temporary impact function Fbm . Then theoptimal liquidation trajectories for the two models coincide i↵✓ vg ⇤ ◆✓ bm ⇤ ◆sA (Yt ) (Y )⇤Gvg= ⇠t = Gbm sA t,t 0,AA
0 (x) and G
where Gvg is the inverse function of x 7! x 2 Fvg
bm is the20
inverse function of x 7! x Fbm (x). We can solve for Fvg to obtain✓◆Z x1 vg
1bm
2 0Fvg (x) =sA
Az Fbm (z) dz.
2 sA0 Az

If Fbm (x) ⇠ x for x small, then Fvg (x) ⇠ x for x small.

Temporary Impact; VG case121086422468101214

Conclusion• We obtain semi-explicit solutions for the optimal liquidation

trajectories when the risk is modelled by a Lévy process.

• We obtain an explicit expression for the connection between

the temporary impact function for the Lévy model and thetemporary impact function for the BM model, such that theoptimal liquidation trajectories are identical.

• There might be a connection between the distribution of thereturns and the temporary impact function, but this would
require further investigations.

References
1 R. Almgren (2003), Optimal execution with nonlinear
impact functions and trading-enhanced risk, Risk 10, 1–18.
2 R. Almgren, C. Thum, E. Hauptmann and H. Li(2005), Direct estimation of equity market impact, Risk 18(7), 58–62.
3 A. Schied, T. Schöneborn and M. Tehranchi (2010),
Optimal basket liquidation for CARA investors isdeterministic, Applied Mathematical Finance 17 (6), 471–489.
4 J. Kallsen and A. Shiryaev (2002), The cumulant
process and Esscher’s change of measure, Finance andStochastics 6, 397–428.

Thank You